New Inequality and Functional for Contact with Friction: The Implicit Standard Material Approach∗

Abstract

This paper is devoted to the analysis of the two- or three-dimensional elastic contact problem with Coulomb friction, quasi-static equilibrium, and small displacements. The classical approach is based on two minimum principles, or variational inequalities: the first for unilateral contact and the second for friction. In practical applications, this leads to an algorithm of alternately solving the two problems until convergence is achieved. A coupled approach using one principle or one inequality only is presented. This new approach, based on a model of material called implicit standard, allows for extension of the notion of a normality law to dissipative behavior with a nonassociated flow rule, such as surface friction. For numerical time integration of the laws, Moreau's implicit method is considered. Nondifferentiable potentials are regularized by means of the augmented Lagrangian technique. A discretized formulation using the finite element method and numerical applications are reported in a separate paper.